5. Chiral Perturbation Theory with HLS

1. 5. Chiral Perturbation Theory with HLS

2. 5.1 Derivative Expansion in the HLS

3. ☆ Expansion Parameter ◎ ordinary ChPT for π chiral symmetry breaking scale ◎ ChPT with HLS ☆ Order Counting ・・・ same as ChPT

4. ２ may cause 1/m corrections ρ ・・・ well-defined limit of m → 0 ρ ☆ Importance of Gauge Invariance ◎ In Matter Field Method ◎ In HLS with R -like gauge fixing ξ gauge invariance

5. 5.2 Lagrangian

6. h ∈ ［SU(N ) ］ f V local ☆ Building blocks ◎ ρ and π fields transform homogeneously

7. Current quark masses can be included ・・・ L, R ; gauge fields of SU(N ) μ μ f L,R ◎ external fields S, P・・・ scalar and pseudoscalar external sources transform homogeneously

8. ☆ Lagrangian at O (p ) 2 F = F at leading order χ π π mass term

9. ☆ Lagrangian at O (p ) 4 4 ◎ Useful Relations → specify independent terms at O(p ) ○ Identities ○ Equations of motions for π, σ, ρ

10. 15 independent terms for N＝ 3 f 9 independent terms for N＝ 2 f ◎ Terms generating vertices with at least 4-legs

11. ＾ ◎ Terms with χ 7 independent terms for N＝ 2 f

12. ＾ ＾ ◎ Terms with V , V or A μν μν μν z , z , z・・・ contribute to 2-point functions 1 2 3

13. 5.3 Quadratic Divergences Importance of quadratic divergence in phase transition

14. Model is defined with cutoff Λ ● ☆ NJL model

15. ◎ Auxiliary field method ; ◎ Effective potential in“chain” approximation

16. = ◎ Stationary condition (Gap equation) self consistency condition

17. ◎ Phase structure ;

18. ◎ Phase change ・・・ triggered by quadratic divergence Phase of bare theory ≠ Phase of quantum theory at bare level ●

19. 5.4 Background field gauge

20. ☆ Background fields background field quantum field background field quantum field

21. ☆ Background fields including external gauge fields

22. ☆ Transformation properties

23. ☆ Gauge fixing and FP ghost three or more quantum fields are included

24. ☆ Lagrangian tree contribution quantum correction at one loop equations of motion for backgroud fiels

25. 5.5 Renormalization Group Equations for HLS Parameters

26. ☆ RGEs for F and z π 2 calculated from A - A two point function μ ν 1-loop contributions quadratic divergence

27. Renormalization

28. ☆ RGEs for F and z π 2 effect of quadratic divergences

29. ☆ RGEs for F and z １ σ calculated from V - V two point function μ ν quadratic divergences

30. calculated from V - V two point function μ ν ☆ RGE for g

31. ☆ RGE for z 3 calculated from V - V two point function μ ν

32. ☆ RGEs for F , a and g π NOTE : (g, a) = (0, 1) ・・・ fixed point

33. ☆ RGEs for z , z and z 2 1 3 4 parameters of O(p ) Lagrangian

34. ☆ RGE for F at μ < m π ρ ρ decouples at μ = m ρ F , g do not run at μ < m ρ σ Fdoes run by π- loop effect π ◎ Physical F π ◎ Effect of finite renormalization

35. 2 ◎ running of F π 2 (86.4MeV) 2 (π) 2 [F (μ)] F (μ) π π ChPT HLS μ 2 0 m 2 ρ

36. ◎ running of a

37. 5.5 Phase Structure of HLS

38. (RGE for F is solved analytically) π ☆ Phase change can occur in the HLS ・ illustration with (g, a) = (0,1) ・・・ fixed point ・ at bare level ・ at quantum level The quantum theory can be in the symmetric phase even if the bare theory is written as if it were in the broken phase.

39. ☆ RGEs ◎ on-shell condition ◎ order parameter

40. ☆ Fixed points (line) ・・・ unphysical

41. ☆ Flow diagram on G = 0 plane symmetric phase VM broken phase

42. ☆ Flow diagram on a = 1 plane symmetric phase VM ρ decoupled broken phase

43. ☆ phase boundary surface

44. ☆ Vector dominance ・ In N = 3 QCD ～ real world f characterized by